If you want large surge currents in a single cell you need fast electron transfer kinetics. Electrically, this means a low internal resistance.
From Wikipedia, the lead-acid design principle is membraneless, meaning that discharge isn't limited by mass transfer kinetics across a membrane (which manifests as internal resistance). The half cell reactions instead both deplete the H2SO4 electrolyte layer, whereby charge balance occurs faster, and manifests a lower initial internal resistance (although it likely increases rapidly during discharge).
Use two different metals, and you would begin getting cross contamination across the electrolyte and lose capacity (in principle at least).
It is also somewhat unique that PbO2 is conductive enough to be used as an electrode, and not just as a deposit on an (inert) electrode (electrical folks feel free to correct me).
I haven't surveyed the periodic table, but I imagine that finding a cheap metal that can do this is rare, nevermind safely.